Abstract

About J. Taylor's regularity result, and minimal sets in R^3

Guy David
University of Paris Sud
Mathematics

Here minimal sets are closed sets $E \i \bf R^3$,
with locally finite Hausdorff (surface) measure $H^2$, and such that for each
Lipschitz function $\varphi : \bf R^3 \to \bf R^3$ such that $W = \{ x \in \bf R^3 ; \varphi(x) \neq x \}$ is bounded,$H^2(E\cap W) \leq H^2(\varphi(E\cap W))$. (Think about infinite soap
films.) Jean Taylor characterized the minimal cones (there are only 3 simple types) and used this to get a good local description of minimal sets, and of a much larger class of almost-minimizers.

We shall try to see whether Taylor's result allows us to show that all minimal sets in $\bf R^3$ are cones. We shall also try to give a
simple account of part of her regularity result, using a Reifenberg-like description.